Lower bounds for high derivatives of smooth functions with given zeros
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866917581232799744 |
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| author | Goldman, Gil Yomdin, Yosef |
| author_facet | Goldman, Gil Yomdin, Yosef |
| contents | Let $f: B^n \rightarrow {\mathbb R}$ be a $d+1$ times continuously differentiable function on the unit ball $B^n$, with $\max_{z\in B^n} |f(z)|=1$. A well-known fact is that if $f$ vanishes on a set $Z\subset B^n$ with a non-empty interior, then for each $k=1,\ldots,d+1$ the norm of the $k$-th derivative $\|f^{(k)}\|$ is at least $M=M(n,k)>0$. A natural question to ask is: What happens for other sets $Z$? In particular, for finite, but sufficiently dense sets?}
This question was partially answered in ([16],[20-22]). This study can be naturally related to a certain special settings of the classical Whitney's smooth extension problem.
Our goal in the present paper is threefold: first, to provide an overview of the relevant questions and existing results in the general Whitney's problem. Second, we provide an overview of our specific setting and some available results. Third, we provide some new results in our direction. These new results extend the recent result of [21], where an answer to the above question is given via the topological information on $Z$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_01388 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Lower bounds for high derivatives of smooth functions with given zeros Goldman, Gil Yomdin, Yosef Classical Analysis and ODEs Let $f: B^n \rightarrow {\mathbb R}$ be a $d+1$ times continuously differentiable function on the unit ball $B^n$, with $\max_{z\in B^n} |f(z)|=1$. A well-known fact is that if $f$ vanishes on a set $Z\subset B^n$ with a non-empty interior, then for each $k=1,\ldots,d+1$ the norm of the $k$-th derivative $\|f^{(k)}\|$ is at least $M=M(n,k)>0$. A natural question to ask is: What happens for other sets $Z$? In particular, for finite, but sufficiently dense sets?} This question was partially answered in ([16],[20-22]). This study can be naturally related to a certain special settings of the classical Whitney's smooth extension problem. Our goal in the present paper is threefold: first, to provide an overview of the relevant questions and existing results in the general Whitney's problem. Second, we provide an overview of our specific setting and some available results. Third, we provide some new results in our direction. These new results extend the recent result of [21], where an answer to the above question is given via the topological information on $Z$. |
| title | Lower bounds for high derivatives of smooth functions with given zeros |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2402.01388 |